Measurement of the triple-differential dijet cross section in proton-proton collisions at s=8TeV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{s}=8\,\text {TeV} $$\end{document} and constraints on parton distribution functions

A measurement is presented of the triple-differential dijet cross section at a centre-of-mass energy of 8TeV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\,\text {TeV}$$\end{document} using 19.7fb-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\,\text {fb}^\text {-1}$$\end{document} of data collected with the CMS detector in proton-proton collisions at the LHC. The cross section is measured as a function of the average transverse momentum, half the rapidity separation, and the boost of the two leading jets in the event. The cross section is corrected for detector effects and compared to calculations in perturbative quantum chromodynamics at next-to-leading order accuracy, complemented with electroweak and nonperturbative corrections. New constraints on parton distribution functions are obtained and the inferred value of the strong coupling constant is αS(MZ)=0.1199±0.0015(exp)-0.0020+0.0031(theo)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha _S(M_\text {Z}) = 0.1199\,\pm {0.0015}\,(\mathrm {exp})\, _{-0.0020}^{+0.0031}\,(\mathrm {theo})$$\end{document}, where MZ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_\text {Z}$$\end{document} is the mass of the Z boson.


Introduction
The pairwise production of hadronic jets is one of the fundamental processes studied at hadron colliders. Dijet events with large transverse momenta can be described by parton-parton scattering in the context of quantum chromodynamics (QCD). Measurements of dijet cross sections can be used to thoroughly test predictions of perturbative QCD (pQCD) at high energies and to constrain parton distribution functions (PDFs). Previous measurements of dijet cross sections in proton-(anti)proton collisions have been performed as a function of dijet mass at the SppS, ISR, and Tevatron colliders [1][2][3][4][5][6]. At the CERN LHC, dijet measurements as a function of dijet mass are reported in Refs. [7][8][9][10][11]. Also, dijet events have been studied triple-differentially in transverse energy and pseudorapidities η 1 and η 2 of the two leading jets [12,13]. e-mail: cms-publication-committee-chair@cern.ch In this paper, a measurement of the triple-differential dijet cross section is presented as a function of the average transverse momentum p T,avg = ( p T,1 + p T,2 )/2 of the two leading jets, half of their rapidity separation y * = |y 1 − y 2 |/2, and the boost of the dijet system y b = |y 1 + y 2 |/2. The dijet event topologies are illustrated in Fig. 1.
The relation between the dijet rapidities and the parton momentum fractions x 1,2 of the incoming protons at leading order (LO) is given by x 1,2 = p T √ s (e ±y 1 + e ±y 2 ), where p T = p T,1 = p T,2 . For large values of y b , the momentum fractions carried by the incoming partons must correspond to one large and one small value, while for small y b the momentum fractions must be approximately equal. In addition, for high transverse momenta of the jets, x values are probed above 0.1, where the proton PDFs are less precisely known.
The decomposition of the dijet cross section into the contributing partonic subprocesses is shown in Fig. 2 at next-to-leading order (NLO) accuracy, obtained using the NLOJet++ program version 4.1.3 [14,15]. At small y b and large p T,avg a significant portion of the cross section corresponds to quark-quark (and small amounts of antiquark-antiquark) scattering with varying shares of equal-or unequaltype quarks. In contrast, for large y b more than 80% of the cross section corresponds to partonic subprocesses with at least one gluon participating in the interaction. As a consequence, new information about the PDFs can be derived from the measurement of the triple-differential dijet cross section.
The data were collected with the CMS detector at √ s = 8 TeV and correspond to an integrated luminosity of 19.7 fb −1 . The measured cross section is corrected for detector effects and is compared to NLO calculations in pQCD, complemented with electroweak (EW) and nonperturbative (NP) corrections. Furthermore, constraints on the PDFs are studied and the strong coupling constant α S (M Z ) is inferred. y * = 1 2 |y 1 − y 2 | y b = 1 2 |y 1 + y 2 | 0 1 2 3 0 1 2 3 Fig. 1 Illustration of the dijet event topologies in the y * and y b kinematic plane. The dijet system can be classified as a same-side or opposite-side jet event according to the boost y b of the two leading jets, thereby providing insight into the parton kinematics

The CMS detector
The central feature of the CMS apparatus is a superconducting solenoid of 6 m internal diameter, providing a magnetic field of 3.8 T. Within the solenoid volume are a silicon pixel and strip tracker, a lead tungstate crystal electromagnetic calorimeter (ECAL), and a brass and scintillator hadron calorimeter (HCAL), each composed of a barrel and two endcap sections. The silicon tracker measures charged particles within the pseudorapidity range |η| < 2.5. It consists of 1440 silicon pixel and 15 148 silicon strip detector modules. The ECAL consists of 75 848 lead tungstate crystals, which provide coverage in pseudorapidity |η| < 1.48 in a barrel region and 1.48 < |η| < 3.0 in two endcap regions. In the region |η| < 1.74, the HCAL cells have widths of 0.087 in pseudorapidity and 0.087 in azimuth (φ). In the η-φ plane, and for |η| < 1. 48, the HCAL cells map on to 5 × 5 arrays of ECAL crystals to form calorimeter towers projecting radially outwards from close to the nominal interaction point. For |η| > 1.74, the coverage of the towers increases progressively to a maximum of 0.174 in Δη and Δφ. Within each tower, the energy deposits in ECAL and HCAL cells are summed to define the calorimeter tower energies, subsequently used to provide the energies and directions of hadronic jets. The forward hadron (HF) calorimeter extends the pseudorapidity coverage provided by the barrel and endcap detectors and uses steel as an absorber and quartz fibers as the sensitive material. The two halves of the HF are located 11.2 m from the interaction region, one on each end, and together they provide coverage in the range 3.0 < |η| < 5.2. Muons are measured in gas-ionisation detectors embedded in the steel flux-return yoke outside the solenoid.
A more detailed description of the CMS detector, together with a definition of the coordinate system used and the relevant kinematic variables, can be found in Ref. [16].

Event reconstruction and selection
Dijet events are collected using five single-jet high-level triggers [17,18], which require at least one jet with p T larger than 80, 140, 200, 260, and 320 GeV, respectively. At trigger level the jets are reconstructed with a simplified version of the particle-flow (PF) event reconstruction described in the following paragraph. All but the highest threshold trigger were prescaled in the 2012 LHC run. The triggers are employed in mutually exclusive regions of the p T,avg spectrum, cf. Table 1, in which their efficiency exceeds 99%.
The PF event algorithm reconstructs and identifies particle candidates with an optimised combination of information from the various elements of the CMS detector [19]. The energy of photons is directly obtained from the ECAL measurement, corrected for zero-suppression effects. The energy of electrons is determined from a combination of the electron momentum at the primary interaction vertex as determined by the tracker, the energy of the corresponding ECAL cluster, and the energy sum of all bremsstrahlung photons spatially compatible with originating from the electron track. The energy of muons is obtained from the curvature of the corresponding track. The energy of charged hadrons is determined from a combination of their momentum measured in the tracker and the matching ECAL and HCAL energy deposits, corrected for zero-suppression effects and for the response function of the calorimeters to hadronic showers. Finally, the energy of neutral hadrons is obtained from the corresponding corrected ECAL and HCAL energies. The leading primary vertex (PV) is chosen as the one with the highest sum of squares of all associated track transverse momenta. The remaining vertices are classified as pileup vertices, which result from additional proton-proton collisions. To reduce the background caused by such additional collisions, charged hadrons within the coverage of the tracker, |η| < 2.5 [20], that unambiguously originate from a pileup vertex are removed.
Hadronic jets are clustered from the reconstructed particles with the infrared-and collinear-safe anti-k T algorithm [21] with a jet size parameter R of 0.7, which is the default for CMS jet measurements. The jet momentum is determined as the vectorial sum of all particle momenta in the jet, and is found in the simulation to be within 5-10% Fig. 2 Relative contributions of all subprocesses to the total cross section at NLO as a function of p T,avg in the various y * and y b bins. The subprocess contributions are grouped into seven categories according to the type of the incoming partons. The calculations have been performed with NLOJet++. The notation implies the sum over initial-state parton flavors as well as interchanged quarks and antiquarks of the true momentum over the whole p T range. Jet energy corrections (JEC) are derived from the simulation, and are confirmed with in situ measurements of the energy balance of dijet, photon+jet, and Z boson+jet events [22,23]. After applying the usual jet energy corrections, a small bias in the reconstructed pseudorapidity of the jets is observed at the edge of the tracker. An additional correction removes this effect.
All events are required to have at least one PV that must be reconstructed from four or more tracks. The longitudinal and transverse distances of the PV to the nominal interaction point of CMS must satisfy |z PV | < 24 cm and ρ PV < 2 cm, respectively. Nonphysical jets are removed by loose jet identification criteria: each jet must contain at least two PF candidates, one of which is a charged hadron, and the jet energy fraction carried by neutral hadrons and photons must be less than 99%. These criteria remove less than 1% of genuine jets.
Only events with at least two jets up to an absolute rapidity of |y| = 5.0 are selected and the two jets leading in p T are required to have transverse momenta greater than 50 GeV and |y| < 3.0. The missing transverse momentum is defined as the negative vector sum of the transverse momenta of all PF candidates in the event. Its magnitude is referred to as p miss T . For consistency with previous jet measurements by CMS, p miss T is required to be smaller than 30% of the scalar sum of the transverse momenta of all PF candidates. For dijet events, which exhibit very little p T imbalance, the impact is practically negligible.

Measurement of the triple-differential dijet cross section
The triple-differential cross section for dijet production is defined as where N denotes the number of dijet events within a given bin, L eff int the effective integrated luminosity, and the product of trigger and event selection efficiencies, which are greater than 99% in the phase space of the measurement. Contributions from background processes, such as tt production, are several orders of magnitude smaller and are neglected. The bin widths are Δp T,avg , Δy * , and Δy b .
The cross section is unfolded to the stable-particle level (lifetime cτ > 1 cm) to correct for detector resolution effects. The iterative D'Agostini algorithm with early stopping [24][25][26], as implemented in the RooUnfold package [27], is employed for the unfolding. The response matrix, which relates the particle-level distribution to the measured distribution at detector level, is derived using a forward smearing technique. An NLOJet++ prediction, obtained with CT14 PDFs [28] and corrected for NP and EW effects, is approximated by a continuous function to represent the distribution at particle level. Subsequently, pseudoevents are distributed uniformly in p T,avg and weighted according to the theoreti-cal prediction. These weighted events are smeared using the jet p T resolution to yield a response matrix and a prediction at detector level. By using large numbers of such pseudoevents, statistical fluctuations in the response matrix are strongly suppressed. The jet energy (or p T ) resolution (JER) is determined from the CMS detector simulation based on the Geant4 toolkit [29] and the pythia 6.4 Monte Carlo (MC) event generator [30] and is corrected for residual differences between data and simulation following Ref. [23]. The rapidity dependence of both the JER from simulation and of the residual differences have been taken into account. The Gaussian p T resolution in the interval |y| < 1 is about 8% at 100 GeV and improves to 5% at 1 TeV. Non-Gaussian tails in the JER, exhibited for jet rapidities close to |y| = 3, are included in a corresponding uncertainty.
The regularisation strength of the iterative unfolding procedure is defined through the number of iterations, whose optimal value is determined by performing a χ 2 test between the original measured data and the unfolded data after smearing with the response matrix. The values obtained for χ 2 per number of degrees of freedom, n dof , in these comparisons approach unity in four iterations and thereafter decrease slowly for additional iterations. The optimal number of iterations is therefore determined to be four. The procedure is in agreement with the criteria of Ref.
[31]. The response matrices derived in this manner for each bin in y * and y b are nearly diagonal. A cross check using the pythia 6 MC event generator as theory and the detector simulation to construct the response matrices revealed no discrepancies compared to the baseline result.
Migrations into and out of the accepted phase space in y * and y b or between bins happen only at a level below 5%. The net effect of these migrations has been included in the respective response matrices and has been cross checked successfully using a 3-dimensional unfolding.
As a consequence of these migrations, small statistical correlations between neighbouring bins of the unfolded cross sections are introduced during the unfolding procedure. The statistical uncertainties after being propagated through the unfolding are smaller than 1% in the majority of the phase space, and amount up to 20% for highest p T,avg .
The dominant systematic uncertainties in the cross section measurement arise from uncertainties in the JEC. Summing up quadratically all JEC uncertainties according to the prescription given in Ref.
[23], the total JEC uncertainty amounts to about 2.5% in the central region and increases to 12% in the forward regions. The 2.6% uncertainty in the integrated luminosity [32] is directly propagated to the cross section. The uncertainty in the JER enters the measurement through the unfolding procedure and results in an additional uncertainty of 1-2% of the unfolded cross section. Non-Gaussian tails in the detector response to jets near |y| = 3.0, the Fig. 3 Overview of all experimental uncertainties affecting the cross section measurement in six bins of y b and y * . The error bars indicate the statistical uncertainty after unfolding. The different lines show the uncertainties resulting from jet energy corrections, jet energy resolu-tion, integrated luminosity, non-Gaussian tails in the resolution, and from residual effects included in the uncorrelated uncertainty. The total uncertainty is obtained by adding all uncertainties in quadrature maximal absolute rapidity considered in this measurement, are responsible for an additional uncertainty of up to 2%. Residual effects of small inefficiencies in the jet identification and trigger selection are covered by an uncorrelated uncertainty of 1% [11]. The total systematic experimental uncertainty ranges from about 3-8% in the central detector region and up to 12% for absolute rapidities near the selection limit of 3.0. Figure 3 depicts all experimental uncertainties as well as the total uncertainty, which is calculated as the quadratic sum of all the contributions from the individual sources.

Theoretical predictions
The NLO predictions for the triple-differential dijet cross section are calculated using NLOJet++ within the framework of fastNLO version 2.1 [33,34]. The renormalisation and factorisation scales μ r and μ f are both set to μ = μ 0 = p T,max · e 0.3y * , a scale choice first investigated in Ref. [35]. The variation of these scales by constant factors as described below is conventionally used to estimate the effect of missing higher orders. The scale uncertainty is reduced in regions with large values of y b with the above-mentioned The NLO QCD correction has been derived with the same NLO PDF in numerator and denominator and is included in the NLO prediction by NLOJet++ choice for μ 0 compared to a prediction with μ 0 = p T,avg . The predictions for cross sections obtained with different central scale choices are compatible within the scale uncertainties. The calculation is performed using the PDF sets CT14, ABM11 [36], MMHT2014 [37], and NNPDF 3.0 [38] at next-to-leading evolution order which are accessed via the LHAPDF 6.1.6 interface [39,40] using the respective values of α S (M Z ) and the supplied α S evolution. The size of the NLO correction is shown in Fig. 4 top left and varies between +10% and +30% at high p T,avg and low y b .
The correction factor c NP k is defined as the ratio between the nominal cross section with and without multiple parton interactions (MPI) and hadronisation (HAD) effects where the superscript indicates the steps in the simulation: the parton shower (PS), the MPI, and the hadronisation. The corresponding correction factor, as displayed in Fig. 4 bottom, is applied in each bin k to the parton-level NLO cross section. It differs from unity by about +10% for lowest p T,avg and becomes negligible above 1 TeV.
To account for differences among the correction factors obtained by using herwig++, pythia 8, and powheg+pythia 8, half of the envelope of all these predictions is taken as the uncertainty and the centre of the envelope is used as the central correction factor.
The contribution from EW effects, which arise mainly from virtual exchanges of massive W and Z bosons, is rel- Fig. 5 Overview of the theoretical uncertainties. The scale uncertainty dominates in the lowp T,avg region. At high p T,avg , and especially in the boosted region, the PDFs become the dominant source of uncertainty evant at high jet p T and central rapidities [49,50]. These corrections, shown in Fig. 4 top right, are smaller than 3% below 1 TeV and reach 8% for the highest p T,avg . Theoretical uncertainties in this correction due to its renormalisation scheme and indirect PDF dependence are considered to be negligible.
The total theoretical uncertainty is obtained as the quadratic sum of NP, scale, and PDF uncertainties. The scale uncertainties are calculated by varying μ r and μ f using multiplicative factors in the following six combi- (1, 2), (2, 1), and (2, 2). The uncertainty is determined as the maximal upwards and downwards variation with respect to the cross section obtained with the nominal scale setting [51,52]. The PDF uncertainties are evaluated according to the NNPDF 3.0 prescription as the standard deviation from the average prediction. Figure 5 shows the relative size of the theoretical uncertainties for the phase-space regions studied. The scale uncertainty dominates in the lowp T,avg region. At high p T,avg , and especially in the boosted region, the PDFs become the dominant source of uncertainty. In total, the theoretical uncertainty increases from about 2% at low p T,avg to at least 10% and up to more than 30% for the highest accessed transverse momenta and rapidities.

Results
The triple-differential dijet cross section is presented in Fig. 6 as a function of p T,avg for six phase-space regions in y * and Fig. 6 The triple-differential dijet cross section in six bins of y * and y b . The data are indicated by different markers for each bin. The theoretical predictions, obtained with NLOJet++ and NNPDF 3.0, and complemented with EW and NP corrections, are depicted by solid lines. Apart from the boosted region, the data are well described by the predictions at NLO accuracy over many orders of magnitude y b . The theoretical predictions are found to be compatible with the unfolded cross section over a wide range of the investigated phase space.
The ratios of the measured cross section to the theoretical predictions from various global PDF sets are shown in Fig. 7. The data are well described by the predictions using the CT14, MMHT 2014, and NNPDF 3.0 PDF sets in most of the analysed phase space. In the boosted regions (y b ≥ 1) differences between data and predictions are observed at high p T,avg , where the less known high-x region of the PDFs is probed. In this boosted dijet topology, the predictions exhibit large PDF uncertainties, as can be seen in Fig. 5. The significantly smaller uncertainties of the data in that region indicate their potential to constrain the PDFs.
Predictions using the ABM 11 PDFs systematically underestimate the data for y b < 2.0. This behavior has been observed previously [53] and can be traced back to a soft gluon PDF accompanied with a low value of α S (M Z ).   Similarly, the c quark mass, set by default to 1.47 GeV, is varied between 1.41 and 1.53 GeV. The minimum Q 2 imposed on the HERA DIS data is set in accordance with the CMS Fig. 8 Ratio of the triple-differential dijet cross section to the NLO-Jet++ prediction using the NNPDF 3.0 set. The data points including statistical uncertainties are indicated by markers, the systematic experimental uncertainty is represented by the hatched band. The solid band shows the PDF, scale, and NP uncertainties quadratically added. The predictions of the NLO MC event generators powheg+pythia 8 and herwig 7 are depicted by solid and dashed lines, respectively inclusive jet analysis described in [53] to Q 2 min = 7.5 GeV 2 , and is varied between Q 2 min = 5.0 GeV 2 and 10.0 GeV 2 . The parameterisation uncertainty is estimated by including additional parameters in the fit, leading to a more flexible functional form of the PDFs. Each parameter is successively added in the PDF fit, and the envelope of all changes to the central PDF fit result is taken as parameterisation uncertainty. The increased flexibility of the PDFs while estimating the parameterisation uncertainty may lead to the seemingly paradoxical effect that, although new data are included, the total uncertainty can increase in regions, where direct con-straints from data are absent. This may happen at very low or at very high x, where the PDF is determined through extrapolation alone. Furthermore, the variation of the starting scale Q 2 0 to 1.6 and 2.2 GeV 2 is considered in this parameterisation uncertainty.
The quality of the resulting PDF fit with and without the dijet measurement is reported in Table 2. The partial χ 2 per data point for each data set as well as the χ 2 /n dof for all data sets demonstrate the compatibility of the CMS dijet measurement and the DIS data from the H1 and ZEUS experiments in a combined fit. Table 2 The partial χ 2 (χ 2 p ) for each data set in the HERA DIS (middle section) or the combined fit including the CMS triple-differential dijet data (right section) are shown. The bottom two lines show the total χ 2 and χ 2 /n dof . The difference between the sum of all χ 2 p and the total χ 2 for the combined fit is attributed to the nuisance parameters  The PDFs obtained for the gluon, u valence, d valence, and sea quarks are presented for a fit with and without the CMS dijet data in Fig. 9 for Q 2 = 10 4 GeV 2 . The uncertainty in the gluon PDF is reduced over a large range in x with the largest impact in the high-x region, where some reduction in uncertainty can also be observed for the valence quark and the sea quark PDFs. For x values beyond ≈ 0.7 or below 10 −3 , the extracted PDFs are not directly constrained by data and should be considered as extrapolations that rely on PDF parameterisation assumptions alone.
The improvement in the uncertainty of the gluon PDF is accompanied by a noticeable change in shape, which is most visible when evolved to low scales as shown in Fig. 10. Compared to the fit with HERA DIS data alone, the gluon PDF shrinks at medium x and increases at high x. A similar effect has been observed before, e.g. in Ref.
[53]. Fig. 10 The gluon PDF as a function of x as derived from HERA inclusive DIS data alone (hatched band) and in combination with CMS dijet data (solid band). The PDF and its total uncertainty are shown at the starting scale Q 2 = 1.9 GeV 2 of the PDF evolution The PDFs are compared in Fig. 11 to those obtained with inclusive jet data at √ s = 8 TeV [61]. The shapes of the PDFs and the uncertainties are similar. Somewhat larger uncertainties in the valence quark distributions are observed in the fit using the dijet data with respect to those obtained from the inclusive jet cross section. This behaviour can be explained by a stronger sensitivity of the dijet data to the light quark distributions, resulting in an increased flexibility of the PDF parameterisation, however, at the cost of an increased uncertainty.
The measurement of the triple-differential dijet cross section not only provides constraints on the PDFs, but also on the strong coupling constant. Therefore, the PDF fit is repeated with an additional free parameter: the strong coupling constant α S (M Z ). The value obtained for the strong coupling constant is  . In contrast to the other CMS results, this analysis is mainly focused on PDF constraints. The running of the strong coupling constant was tested only indirectly via the renormalisation group equations. No explicit test of the running was carried out by subdividing the phase space into regions corresponding to different values of the renormalisation scale.

Summary
A measurement of the triple-differential dijet cross section is presented for √ s = 8 TeV. The data are found to be well described by NLO predictions corrected for nonperturbative and electroweak effects, except for highly boosted event topologies that suffer from large uncertainties in parton distribution functions (PDFs).
The precise data constrain the PDFs, especially in the highly boosted regime that probes the highest fractions x of the proton momentum carried by a parton. The impact of the data on the PDFs is demonstrated by performing a simultaneous fit to cross sections of deep-inelastic scattering obtained by the HERA experiments and the dijet cross section measured in this analysis. When including the dijet data, an increased gluon PDF at high x is obtained and the overall uncertainties of the PDFs, especially those of the gluon distribution, are significantly reduced. In contrast to a fit that uses inclusive jet data, this measurement carries more information on the valence-quark content of the proton such that a more flexible parameterisation is needed to describe the low-x behaviour of the u and d valence quark PDFs. This higher sensitivity is accompanied by slightly larger uncertainties in the valence quark distributions as a consequence of the greater flexibility in the parameterisation of the PDFs.
In a simultaneous fit the strong coupling constant α S (M Z ) is extracted together with the PDFs. The value obtained at the mass of the Z boson is